On steady solutions of symmetry-preserving perturbations of the two-dimensional Couette flow problem

Abstract

We discuss homotopy methods for obtaining two-dimensional (2D) steady solutions of the planar Couette flow problem together with the relationship of these solutions to the spectrum of its linearization and to the symmetry of the nonlinear problem. In order to obtain these solutions, we classify the near-degeneracy of the real part of the spectrum, construct a certain homotopy that preserves the symmetry of the nonlinear problem, and propose fixed-point space computations to compute these solutions. We obtain numerical solutions for extremely small symmetry-preserving perturbations of the nonlinear equations governing 2D Couette flow.